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CHAPTER 1
SPECIAL RELATIVITY
To understand black holes, we have to learn some relativity. The theory of relativity is split into two parts: special and general. Albert Einstein came up with the special theory of relativity in 1905. It deals with objects moving relative to one another, and with the way an observer's experience of space and time depends on how she is moving. The central ideas of special relativity can be formulated in geometrical terms using a beautiful concept called Minkowski spacetime.
General relativity subsumes special relativity and also includes gravity. General relativity is the theory we need in order to really understand black holes. Einstein developed general relativity over a period of years, culminating in a paper in late 1915 in which he presented the so-called Einstein field equations. These equations describe how gravity distorts Minkowski spacetime into a curved spacetime geometry, for example the Schwarzschild black hole geometry that we will describe in Chapter 3. Special relativity is simpler and easier than general relativity because gravity is neglected — that is, gravity is ignored, or presumed to be too weak an effect to be significant.
Special relativity includes the formula E = mc2, relating energy E, mass m, and the speed of light c. This is one of the most famous equations in all of physics, possibly in all of human understanding. E = mc2 made it possible to foresee the awesome power of nuclear weapons, and it is at the core of our hopes, as yet unrealized, for a clean source of energy from nuclear fusion. E = mc2 is also very relevant to black hole physics. For example, the 3 solar masses' worth of energy ejected from the first observed black hole collision is a prime illustration of the equivalence of mass and energy. To get an idea of just how cataclysmic this collision was, consider that the mass converted into energy in the explosion of a nuclear weapon (assuming a yield of 400 kilotons) is a mere 19 grams.
Special relativity is closely related to James Clerk Maxwell's theory of electromagnetism. Indeed, an early hint of the relativistic view of space and time emerged in the late 1800s in the form of so-called Lorentz transformations, which explain how observers' perceptions of electromagnetic phenomena depend on how the observers are moving. The most familiar electromagnetic phenomenon is light, which is a traveling wave of electric and magnetic fields. A consequence of Maxwell's theory is that light has a definite speed. Relativity is built around the idea that this speed is truly a constant, independent of the motion of the observer.
The motion of observers is described in special relativity in terms of frames of reference. To get an idea of what a frame of reference is, think of a high-speed train. If all the passengers are seated and all the luggage is stowed, then everything on the train indeed is stationary with respect to the train itself. But the train is moving quickly relative to the Earth. Let's assume that the train is moving in a straight line at constant speed. To give a fully precise account of frames of reference, we should stipulate the absence of any significant gravitational field. For example, instead of a train running at constant speed along the Earth's surface, we would need to consider a spaceship coasting at constant speed in otherwise empty space. Earth's gravitational field is weak enough that, for present purposes, we can ignore its effects on the train and work with just the special theory of relativity rather than the general theory.
Without looking out the windows, it's hard to tell how fast the train is moving. In a situation where the train has fantastic suspension and the track is very even, and where the blinds on all the windows are down, it would be impossible to know that the train is moving at all. The train provides a frame of reference — the one that its passengers naturally use to judge whether something inside the train is moving. They can't tell (in the ideal situation just described) whether the entire train is moving. But they certainly know when someone walks up the aisle, because such a person is moving relative to their frame of reference. Furthermore, all physical phenomena, like balls dropping or tops spinning, would behave the same, as observed by an observer on the train, whether the train is actually moving or not. Briefly then, a frame of reference is a way of looking at space and time which is associated with an observer, or a group of observers, in a state of uniform motion. Uniform motion means that the train is not speeding up, or slowing down, or turning. If the train is doing one of these things, then the passengers will notice; for example, rapid acceleration pushes them back in their seats, whereas rapid deceleration throws them forward.
Let's imagine our train passing through a station without stopping or slowing down. The passengers on the train, call them Alice, Allan, and Avery, are observers in a moving frame of reference which we'll call the A-frame. Meanwhile, their friends Bob, Betsy, and Bill stand on the platform, in a stationary frame of reference which we'll call the B-frame. To draw these frames of reference, we put B-frame position on the horizontal axis and B-frame time on the vertical axis, and we map out the trajectories of our various observers through space and time, so that over time, B-frame observers always stay at the same B-frame positions, whereas A-frame observers move forward. The resulting diagram is actually Minkowski spacetime! The word spacetime refers to the fact that we are showing space and time on the same diagram. It's possible to take a different perspective on Minkowski spacetime, such that A-frame observers are shown as stationary while B-frame observers move backward. More on that perspective later.
Special relativity hinges on the assumption that the speed of light is constant. In other words, the speed of light is supposed to be the same when measured by the observers on the train as when measured by the observers on the platform. If that weren't so, then by measuring the speed of light, an observer could tell which of the two frames of reference she was in. But a core tenet in relativity theory is that physics should be the same in any frame of reference, so that you really can't tell which frame you're in through any physical measurement. According to this tenet, we cannot pick out a frame and say, "Remaining in this frame is what it means to be stationary. Motion consists of being in a different frame." We can only say, "Any frame is as good as any other. The only idea of motion that we can permit is motion of one observer with respect to another." In other words, states of motion are not absolute; they are relative. Thus it was a misnomer to refer to the A-frame as moving and the B-frame as stationary. All we can really say is that they are moving with respect to one another. (The idea of the B-frame being stationary seemed natural, though, because we were implicitly thinking of motion relative to the Earth.)
The intuition we've explained about relative motion seems like common sense, and we should ask ourselves how we can possibly get any leverage from it on questions relating to the deep nature of space and time. The key ingredient is Maxwell's theory of electromagnetism. What this theory tells us (among other things) is that if Alice pulls out a laser pointer and sends a pulse of light forward, toward the front of the train, and Bob does the same, then the two light pulses travel forward at exactly the same velocity. This seems like another innocuous claim, but it's not! For example, if we arrange for the train to go at 99% of the speed of light (so obviously not an American train), then wouldn't Bob measure a laser pulse shot forward by Alice to be traveling at almost double the speed of light? After all, she is moving forward at 99% of the speed of light relative to Bob, and her light pulse moves forward at the speed of light relative to her, so it seems like Bob should measure her light pulse to be moving forward at 199% of the speed of light. But according to electromagnetism, he doesn't! He measures it to be moving at precisely the same speed of light, relative to him, that Alice would report if she measured its motion relative to her.
How is this possible? The answer is that Alice and Bob measure the passage of time differently, and they also measure length differently. The details of how this happens are encoded in the Lorentz transformation, which is a mathematical expression relating time and length in the A-frame to time and length in the B-frame. A Lorentz transformation is easy to draw using Minkowski spacetime. Before the Lorentz transformation (the left side of Figure 1.1), we can think of the B-frame as stationary and the A-frame as moving forward. After the Lorentz transformation (the right side of Figure 1.1), the A-frame is stationary and the B-frame is moving backward! A Lorentz transformation is just the change of perspective between the account that Bob would offer based on thinking of his frame as stationary, and the one that Alice would offer based on thinking of her frame as stationary.
Key consequences of the Lorentz transformation include time dilation and length contraction. We're going to explain time dilation first because it's easier to describe. Suppose that at noon on Friday you get on a train at Princeton Junction. For convenience, we're going to say that this time and place correspond to the origin of Minkowski space, where the t and x axes cross. Now, there are fast trains and slow trains that go through Princeton Junction; some go north toward New York, and some go south toward Philadelphia; and you can decide which one you want. What you're going to do is ride the train for exactly one hour by your watch, and then get off and mark where you end up. Obviously, if you take a fast train, you get farther. But beware of the assumption that you get exactly twice as far riding a train that goes twice as fast. The tricky part is that you're riding the train for exactly one hour as measured by your own watch. The speed of a train is something observers who are stationary relative to the ground would measure, and their watches run a bit differently from yours because they're in a different frame of reference.
So where do you end up? More generally, if you and a bunch of friends all take different trains (all departing Princeton Junction at the same time), where do you all wind up? The answer is that you all wind up somewhere on a hyperbola in Minkowski spacetime (see figure 1.2). In other words, the hyperbola is the set of all possible final locations that you can reach after precisely one hour of your own travel time. One possible final location is Princeton Junction itself, at precisely 1 p.m. Princeton Junction time. The way you wind up there after an hour is if you are silly enough to spend an hour on a train that doesn't move at all. In that situation, of course it's 1 p.m. Princeton Junction time when you "arrive," because your frame of reference is the same as the station's, so that your watch exactly keeps pace with station time. If instead you get on a train that actually goes somewhere, your watch runs slower than station time, so when you get off after an hour of perceived travel time, station time is actually later than you think it ought to be. This later-than-you-think effect, known as time dilation, is captured in Minkowski spacetime by the way the hyperbola curves upward in the time direction as you go to locations farther and farther from your starting point. Minkowski spacetime is sometimes called hyperbolic geometry, in reference to precisely the type of hyperbola we have been discussing.
In Minkowski spacetime, we visualize the constant speed of light by drawing light rays at precisely a 45° angle relative to the vertical time axis. You'll notice that the hyperbola of possible endpoints for a one-hour train ride is wholly within the region of spacetime between two light rays emanating from the origin. This is the way Minkowski spacetime encodes the statement that none of our trains can go faster than light.
It may seem like our discussion of time dilation doesn't have much to do with Lorentz transformations. To see that it really does, let's go back to calling the reference frame of the train the A-frame, while the reference frame of the Earth is the B-frame. Suppose Alice spends an hour in the A-frame on her way from Princeton Junction to New York. Meanwhile, Bob and his friends remain stationary with respect to the Earth. How should they figure out the time of Alice's arrival? It's not very useful for her to call them when she arrives, because the signal she would use could only travel at the speed of light, and Bob and friends would have to do a calculation based on the time they received her call, the speed of the signal, and the distance to New York City to figure out when Alice arrived. That all sounds too tricky. So Bob figures out a better way. He synchronizes his watch with one of his friends, let's say Bill, and Bob and Bill take up positions at the Princeton Junction and New York City train stations, respectively. Bob measures when Alice leaves, and Bill measures when she arrives. No telephony required. It might seem tricky to synchronize watches in a reliable way between distant observers, but one good strategy would be for Bob and Bill both to start out halfway between Princeton Junction and New York City, synchronize their watches while standing next to one another, and then walk at identical speeds to their respective stations, all well before Alice boards her train.
In this whole narrative of Alice's train ride, the A-frame is clearly privileged, because Alice doesn't need any friends to figure out the duration of her train ride, whereas Bob and Bill must cooperate to make their measurement of the time. The time interval that Alice measures is called proper time because she measures it while remaining at a fixed location in her own frame of reference (the A-frame). The time interval that Bob and Bill measure is dilated time, which always must be greater than proper time. Dilated time is part of how the A-frame and B-frame perspectives on spacetime are related. The Lorentz transformation between the A-frame and the B-frame contains time dilation, and more.
A similar discussion can be used to describe length contraction. Instead of a train ride, let's imagine that Bob, Bill, and Alice go to the Olympics, where Alice hopes to set a record in pole-vaulting. Her secret is that she can run really fast, at 87% of the speed of light. (For some reason, she leaves the 100 meter dash to Usain Bolt, even though she figures she could post a time of under 0.4 microseconds.) Alice chooses a 6 meter pole, which is longer than most vaulters want, but after all she is pretty exceptional. Bob and Bill don't believe that Alice is using a pole that long, so they resolve to measure it as Alice charges down the track, holding her pole perfectly horizontal as she goes. Clearly, they've got a tough job. How can they actually make the measurement? Here is what they come up with. First, they synchronize their watches. Then they stand somewhat less than 6 meters apart, and they agree that at precisely the same time, they're going to glance up at Alice and record which part of her pole they see. After many attempts, they manage to arrange themselves so that Bob sees the tail end of Alice's pole, while Bill sees the front tip. Then they measure the distance between themselves. The answer is that they are only 3 meters apart. They reasonably conclude that Alice's pole is 3 meters long. They approach Alice and explain what they found. Alice protests that they can't have gotten it right. She enlists the help of her two friends, Allan and Avery, who run with her (apparently they're equally good sprinters) and measure her pole in her frame. The answer they find is that her pole is 6 meters long.
Once again, the A-frame is privileged in this discussion, because it's the frame in which Alice's pole is stationary. Its length as measured in the A-frame is called the proper length. Its length as measured in the B-frame is always shorter, and it is termed the contracted length. Time dilation and length contraction are closely linked, as we can appreciate in this example by considering what Alice would say about her experience running down the track toward the bar. As measured in her frame, it takes her half as long to get to the bar as Bob and Bill would have measured by the protocol we discussed above in reference to Alice's train ride to New York City. Time dilation, then, involves a factor of two for Alice's record-busting sprint at 87% of the speed of light. Length contraction also involves a factor of two: A-frame observers say her pole is 6 meters long, and B-frame observers say it is 3 meters long. In general, time dilation and length contraction always involve the same factor, sometimes called the Lorentz factor.
There seems to be a disconnect between our discussion of special relativity, which focuses on spacetime geometry, and the famous equation E = mc2. Let's try to bridge this gap by considering a partial derivation of E = mc2, where the most important steps can be illustrated geometrically. Our argument is only a partial derivation because it will involve some approximations and a couple of other formulas which we don't fully justify or derive.
(Continues…)
Excerpted from "The Little Book of Black Holes"
by .
Copyright © 2017 Steven S. Gubser and Frans Pretorius.
Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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